I have a question:

We all agree, that only Sith use red lightsabres.
Do we agree, that Sith only use red lightsabres?

If we do so: Is Anakin in EP3 not a Sith yet? Because on Mustafar, in the battle with Obi-Wan, Anakin still wields his blue sabre, the one, Obi-Wan hands over to Luke in EP4...

Greetings,
ikarus

He has to make his red lightsabre, and hadn't had time to do so, so he just used his blue one!

Oh hai Ikarus!
Actually, you have to think of it this way:
1. Anakin was in the same room watching the battle of Windu and Sidious. At the time Windu died, Anakin pledges allegiance to the Sith.
2. The timing was short as Sidious is the puppeteer of the whole Jedi and the Clone Wars. Sidious orders 2 tasks in quick succession to end the war, so he can be extreme ruler. Anakin uses his lightsaber (the one he made as a Jedi) to kill other Jedi, and then again on Mustafar.
3. Anakin loses his lightsaber at Mustafar as he was 'dying'. This goes to Luke in ANH.

It's interesting to see if Anakin would have still kept his blue lightsaber had he not lost it, or changed crystal to red...

IIRC, Obi-Wan reaches down and picks up the lightsaber as Anakin is trying to crawl out of the pit (after Obi-Wan has cut off a leg and an arm, or something like that). So Anakin/Vader would have HAD to have built a new saber anyway.

I think he would have built an entirely new saber anyway, had Obi-Wan not taken that one, because aren't the red crystals more powerful or something, even though they are not as stable?